Maximum-Length Low-Density MDS Codes and Near Resolvable Designs

We study the relation between near resolvable designs and ${\mathbb{F}_q}$ -linear codes over $\mathbb{F}_q^b$. We use the incidence matrix of a near resolvable design to construct a parity-check matrix of an ${\mathbb{F}_q}$-linear code. We show an equivalence between the construction of a new class of near resolvable designs NRB(rb + 1, r), that we call r-complete, and the well-known ${\mathbb{F}_q}$-linear codes over $\mathbb{F}_q^b$ with length n and dimension n – r which have the following good properties: (i) they are maximum distance separable (MDS), (ii) they are low-density, and (iii) they reach the maximum length of any MDS lowest density code.